Flat bands, non-trivial band topology and rotation symmetry breaking in layered kagome-lattice RbTi3Bi5

A representative class of kagome materials, AV3Sb5 (A = K, Rb, Cs), hosts several unconventional phases such as superconductivity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 non-trivial topological states, and electronic nematic states. These can often coexist with intertwined charge-density wave states. Recently, the discovery of the isostructural titanium-based single-crystals, ATi3Bi5 (A = K, Rb, Cs), which exhibit similar multiple exotic states but without the concomitant charge-density wave, has opened an opportunity to disentangle these complex states in kagome lattices. Here, we combine high-resolution angle-resolved photoemission spectroscopy and first-principles calculations to investigate the low-lying electronic structure of RbTi3Bi5. We demonstrate the coexistence of flat bands and several non-trivial states, including type-II Dirac nodal lines and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 non-trivial topological surface states. Our findings also provide evidence for rotational symmetry breaking in RbTi3Bi5, suggesting a directionality to the electronic structure and the possible emergence of pure electronic nematicity in this family of kagome compounds.

Supplementary Notes 1-6 Supplementary Note 1. Characterization of single crystal. RbTi 3 Bi 5 single crystals have a typical size of 3 × 5 × 0.5 mm 3 [Fig. S1(a)], all the sample needs to prepare in an argon-filled glove box to prevent the oxidation and hydrolysis of Rb. Single-crystal XRD was collected using a Bruker D8 x-ray diffractometer with Cu K α radiation (λ= 0.15418 nm) at room temperature. The XRD pattern of a RbTi 3 Bi 5 single crystal [ Supplementary Figure 1 (b)] reveals that the crystal surface is parallel to the (00l)-plane. The estimated c-axial lattice constant is about 9.065Å, close to the previously reported values. We conducted core-level spectroscopy with the photon energy of 120 eV. In Supplementary Fig. 1(c), we present the core-level photoemission intensity plot of RbTi 3 Bi 5 where the characteristic peaks of Rb-4s/4p, Ti-3s/3p, and Bi-5d orbitals are clearly observed, with no extra peaks observed, suggesting the phase pure RbTi 3 Bi 5 single crystals.
Supplementary Note 2. Overview of band structures. We performed the measurements of the constant energy contours from Fermi Level (E F ) to E F -1.0 eV with 66 eV photon, as shown in Supplementary Fig. 2. It shows typical band structure of kagome patterns, which consists of a circle-like electron pocket (α band), a hexagonal-like electron pocket (β band) around Γ point, another larger hexagonal-like electron pocket (γ band) encircle β band and with 30 • respect to it is also been observed, this γ band then cross Fermi surface again around K point, and constitutes a triangle-like electron pocket around K point. In contrast, there is a rhombic hole pocket (δ bands) around M points. The orbital character of the highlighted bands is presented in the Supplementary Fig. 3. The Fermi surface are mainly constituted by d-orbits of Ti-atoms and p-orbits of Bi atom, the calculated orbital distribution at Fermi surface is presented in Supplementary Fig. 3(b), and more detailed orbital calculations are presented in Supplementary Fig.  3(d)-(e). According to the calculation, we can summarize the orbital characters of different bands in Supplementary  Fig. 3(c), where the inner circular pocket (red) around Γ point mainly originates from the Bi p z orbital, the hexagonlike one (green) is from Ti d xz /d yz orbitals, the another hexagon-like pocket (yellow) is mainly from Ti d xy /d x 2 −y 2 orbitals, and the rhombic-like and triangle-like bands (gray) are mainly attributed to Ti p x /p y orbitals.
Supplementary Note 3. Features of flat bands. We inspected the intensity plots and corresponding second derivative plots of RbTi 3 Bi 5 , CsTi 3 Bi 5 , and KTi 3 Bi 5 with 66, 86, and 74 eV photons, respectively, along high symmetry-paths, which correspond to M -K-Γ-K-M lines, as shown in Supplementary Figure 4. This flat band seemingly spans the entire BZ and is distinct from our calculation, in which the flat band is absent in the regions along Γ − K and Γ − M , as shown in Figure 1 The origin of such a shadow flat band in ATi 3 Bi 5 should be closely linked with the small flat band. In this way, we conjecture that there might exist several possible origins of this shadow flat band. Firstly, we speculate that it might originate from some rather localized states in ATi 3 Bi 5 , e.g., parts of localized titanium 3d electrons due to spin frustration or impurity states with relatively lower intensity and dispersionless feature. Secondly, this feature might be related to the k z broadening of electronic states at Γ. Although the in-plane electron hopping is confined in kagome lattices, the interlayer electron hopping is still significant in real three-dimensional stacked kagome lattices, resulting in dispersive bands along the k z direction. Our DFT calculations have indicated that the band structure around Γ is largely dispersive along k z , which would result in a band structure evolution from the electron-like to "drumhead"like, as shown by the highlighted area around Γ/A in Supplementary Figure 6. Since our ARPES measurements were performed mainly using VUV photons, the mean escape lengths of resulting photoelectrons are limited. Consequently, the photoemission spectra may be more influenced by the k z broadening effect. However, we note that the origin of this shadow flat band needs more research to pin down.
Supplementary Note 4. Temperature dependent measurements. We performed temperature-dependent ARPES experiments on RbTi 3 Bi 5 , and our results further verified the absence of charge density waves. By comparing the band dispersions taken along high-symmetry direction at low temperature (11 K) and high temperatures (200 K), as shown in Supplementary Figure 9, we find all the band features does not show any changes.
Supplementary Note 5. Autocorrelation of ARPES spectra (AC-ARPES). The original data are extracted from constant energies counters extracted from different binding energies integrated with the spectral weight over E F ± 25 meV. The AC-ARPES in Fig. 5c-e and Supplementary Fig. 11-14 are calculated from the ARPES intensity maps with following equation where A(k, E) is the spectral function at binding energy E at k points in the BZ, and α is constant [1][2][3][4]. In Supplementary Figure 11, we compared the AC-ARPES spectra extracted from E F of RbTi 3 Bi 5 and KV 3 Sb 5 , both of them shows anisotropic intensity distribution. We have also compared the AC-ARPES extracted from E F to E F -0.5 eV by step of 0.1 eV, as shown in Supplementary Fig. 13. It shows hint of anisotropic intensity distribution near E F , but on higher binding energies, the six-fold symmetry recovered.
In Supplementary Fig. 12a, we identified the scattering vectors according to the ARPES Fermi surface of RbTi 3 Bi 5 (q 1 q 4 ), and we append them on the autocorrealtion map, as shown in Supplementary Fig. 12b. As the band structures on Fermi surface are complex, to distinguish the different scattering routes, we further extract the main bands from the models and redo the autocorrelation simulation, as shown in Supplementary Fig. 12c to 12f. The scattering vectors (q 1 , q ′ 1 , q ′′ 1 ) origin from the larger hexagon pocket around Γ point primarily give rise to a bright line surrounding the Γ point, as shown in Supplementary Fig. 12c; the scattering vectors (q 2 , q ′ 2 ) origin from the smaller hexagon pocket around Γ point primarily give rise to the bright spots around K point, as shown in Supplementary  Fig. 12d; the scattering vectors (q 3 ) between α band around Γ point and δ bands around M point gives rise to the bright spots near M points, as shown in Supplementary Fig. 12e; the scattering vectors (q 4 ) between those triangle and rhombic-like pockets on the edges of Brillouin zone mainly contribute to the three parallel lines along Γ − M directions, as shown in Supplementary Fig. 12f.
Additionally, we have also demonstrated the AC-ARPES at 200 K in Supplementary Figure 14, which is similar to the data take at low temperature [ Fig. 5c-d].
Meanwhile, we also attempted to measure rotated samples, as shown in Supplementary Figure 15. When the sample was rotated by -15 • and -30 • , both corresponding autocorrelation results rotate along with the sample. In particular, the characteristic feature of anisotropy in the amplitudes between the three nominally identical directions keeps rotating by the same degrees, as shown in Supplementary Figure 15. We extract the intensity distribution curves along three different Γ − M directions in Supplementary Figure 15c. We observed a noticeable intensity peak at the position indicated by q 3 in two out of the three directions (yellow and blue dashed cut lines). While the spectra weight on the three parallel lines appeared to be fixed during sample rotation, influenced by the polarization effect, the intensity peaks associated with q 3 still rotated with the sample. These q 3 peaks were distinguishable from the yellow and blue cut lines but indistinguishable from the red one in -15 • and -30 • rotated samples, as depicted in Supplementary Fig. 15(f) and 15(i). This result unambiguously indicates that the matrix element effects should not dominate the anisotropy in the autocorrelation. We speculate that this may be attributed to a small distortion of the δ-bands, which are primarily contributed by Bi p x,y orbitals and hybrid orbitals of Bi p x,y and Ti d xy orbitals. These bands could undergo deformation in nematic phases. However, due to the complexity of Fermi surface scattering and the small magnitude of the distortion, we were unable to fully pin down the exact origin of the rotation symmetry breaking, which still wait for further exploration.
To validate the feasibility of autocorrealtion computational approach in "135" series kagome materials, we conducted a similar investigation on KV 3 Sb 5 [5, 6], a known kagome metal with nematic phase. In Supplementary Fig. 16(a), we have marked the different scattering channels by double-head arrows along different Γ − M directions: q 1 represents the scattering from the circular-like band around Γ to the folded circular-like band around M , q 2 represents that from the circular-like band around Γ to the "bow"-like band around M . The autocorrelation calculation enables us to transfer the k-space spectrum into q-space, as demonstrated in Supplementary Fig. 16(b). Since the C 6 rotation symmetry is broken in KV 3 Sb 5 , the band structure of KV 3 Sb 5 also exhibits an anisotropic feature. To highlight the anisotropic feature in the q-space image of autocorrelation, we directly compared the three intensity curves taken along Cut-1, Cut-2 and Cut-3, as shown in Supplementary Fig. 16(c). We observed that the intensity at q 2 along all three directions are approximately equal, while the adjacent q 1 peak along Cut-3 is much higher than those along Cut-1 and Cut-2. Moreover, the anisotropy feature can be more clearly seen from the autocorrelation result of the constant energy intensity map at E F -0.2 eV, as illustrated in Supplementary Fig. 15(d)-(f). We note that the intensity of q 2 peaks is nearly equal for all three directions. However, the q 1 peak is rather outstanding along Cut-3 while it is evidently suppressed along both Cut-1 and Cut-2. Both bulges at q 1 taken along Cut-3 for E F and E F -0.2 eV reflect relatively stronger correlation between the pristine and CDW folded bands along q 1 for KV 3 Sb 5 , unambiguously indicative of the breaking of original C 6 symmetry therein.
Supplementary Note 6. Calculation of topological index. The Z 2 topological invariant is computed via the parity products at the time reversal invariant momentum (TRIM) points. Although the inversion symmetry compels the 122 and 124 bands to cross along Γ − A, thereby inhibiting the formation of a continuous gap between these two bands, this criterion can still be applied here according to the theory of symmetry indicators [7,8]. According to this theory, parity eigenvalues at the TRIM points can be used to identify band topology, irrespective of the presence or absence of a continuous band gap. For ATi 3 Bi 5 , which exhibits both time-reversal and space-inversion symmetries, the following general form expresses its topological invariant: where the three Z 2 indices correspond to the three weak indices of the Fu-Kane criterion and the strong Z 2 index is elevated to a Z 4 factor. One can still utilize the Z 2 index to determine whether the compound is topological or not. However, to discern whether the compound is an insulator or semimetal, we concur that further analysis of the band crossings is necessary. Supplementary Figures 1-11 Supplementary